Finite-sample non-degenerate e-variable for mean-zero distributions with unknown variance

Construct a finite-sample, non-degenerate e-variable for the composite null P = { distributions on R with E[X] = 0 and Var(X) ∈ (0,∞) } that does not rely on a known bound on the variance. The e-variable must satisfy E^P[E] ≤ 1 for every P in this class, and must not collapse to a constant (degenerate) value.

Background

The authors present an asymptotic e-variable for testing mean-zero under unknown variance, exploiting asymptotic normality of S_n/V_n. They remark that non-asymptotic e-variables exist when a variance bound is available, but note a gap in the finite-sample theory for the case of unknown (and unbounded) variance.

This highlights a specific unresolved construction problem: to design a finite-sample e-variable that is valid for all distributions with mean zero and finite (but unknown) variance, and that is not trivially degenerate.